Inverses include: multiplicative inverse of a number (reciprocal), inverse function, matrix inverse, etc. A subject tag such as (linear-algebra), (algebra-precalculus) or (arithmetic) should be added to clarify in which sense "inverse" is used. This tag should never be the only tag on a question.

learn more… | top users | synonyms

1
vote
2answers
36 views

Solving $3\times 3$ matrix equations:

I am familiar with finding the inverse of matrices, but struggle to formulate matrix equations. In this particular question, one is asked to find the elementary matrix E where $E*A = B$. $A$ is ...
0
votes
2answers
48 views

What kind of distribution in this chart?

Could you tell me what kind of distribution is this? Chart This is the data: ...
11
votes
3answers
213 views

First order differential equation involving inverses

My question is to find the solutions to the following $\frac{df(x)}{dx} = f^{-1} (x)$ where $f^{-1} (x)$ refers to the inverse of the function f. The domain really isn't important, though I am ...
1
vote
1answer
33 views

sum of matrix inverse problem

Recently, when I was reading matrix analysis, a formula confused me a lot: If $A+B$ is nonsingular, then the following is true, $$A(A + B)^{-1}B = B(A + B)^{-1}A$$ I tested some random ...
0
votes
1answer
34 views

Logarithmic to linear

Given this function: $$\frac{1.0}{1024.0} + \frac{x}{100.0} * \frac{1023.0}{1024.0} = y$$ $$10 * \frac{\log_{10}(y)}{\log_{10}(2)} = z$$ $$z * 100 = a$$ ...
1
vote
1answer
53 views

Relationship between inverse of related matrices

Suppose I have a matrix $A \in \mathbb{R}^{m\times n}$ with $m \geq n$ and suppose that a matrix $G=(A^T A)^{-1}$ exists. Now suppose that I have an other matrix $B \in \mathbb{R}^{m\times m}$ that ...
-2
votes
1answer
21 views

Is this system invertible?

$y(t) = \int\limits_{-\infty}^{\infty} \frac {x(t)^2}{x(t-1)} dt\\$ I was trying to prove or disprove the invertibility of this function. The only thing I could think of was differentiating it. But ...
2
votes
2answers
40 views

Compute the indicated power of a matrix

Compute the indicated power of the matrix: $A^8$ $ A = \begin{bmatrix}2&1&2\\2&1&2\\2&1&2\end{bmatrix} $ I calculated the eigenvalues: $ \lambda_1 = \lambda_2 = 0, \lambda_3 ...
0
votes
1answer
19 views

Trace of Hermitian Positive Semidefinite Matrix

Well, the question I want to ask is as follows. Suppose A and B are Hermitian Positive Semidefinite (PSD) matrices, I wonder if it is possible to prove $Tr(A*(A+B)^{-1})\in (0,1]$ (if it is ...
0
votes
2answers
53 views

Pseudo-eigenvector times matrix inverse

Actually I don't know what should be a good title of my question. Here comes the simplified version of the question. Let's call it case 1. As we know, for a non-singular matrix $\textbf{A}$ with ...
1
vote
0answers
31 views

Confusion regarding logic in paper, “A NOTE ON THE INVERSION OF POWER SERIES,” published in the AMS journal

I was reading "A NOTE ON THE INVERSION OF POWER SERIES" and was able to follow the paper's reasoning until the bottom of the second page, where it states: in fact we can calculate the power series ...
0
votes
1answer
47 views

Using Chinese Remainder Theorem to find an integer $x$ for which $ x\equiv 3\pmod 4 x\equiv 5\pmod 9 x\equiv 10\pmod {35} $

Hello I have got problems with understanding the reduction method in CRT. We have got system like this $$x\equiv 3\pmod 4$$ $$x\equiv 5\pmod 9$$ $$x\equiv 10\pmod {35}$$ There is a way to do this ...
2
votes
0answers
43 views

'Stable' Ways To Invert A Matrix

So lets say that I need to invert a matrix that is generally dense and is poorly conditioned. What are some ways I can get an accurate inverse? Here are my candidates: SVD Inverse Inverse Via ...
0
votes
2answers
64 views

If a matrix $A^2$ is invertible, is $A^3$ invertible? [closed]

I know how to find out if a matrix $A^2$ is invertible if $A^3$ is invertible, but how can you find out invertibility if it's the the other way around?
0
votes
2answers
50 views

Solving for $x$ in a Laplace equation

So I have this Laplace equation: $$s^{2}x+4sx+5=\frac{s}{s-1}$$ And I want to solve for $x$. My result is the following: $$x = \frac{5-4s}{s^{3}+3s^{2}-4s}$$ Which is also the same answer that for ...
0
votes
2answers
63 views

Finding all left inverses of a matrix

I have to find all left inverses of a matrix $A = \begin{bmatrix} 2&-1 \\ 5 & 3\\ -2& 1 \end{bmatrix}$ I created a matrix to the left of $A$, $\begin{bmatrix} a &b &c \\ ...
2
votes
3answers
64 views

Necessary and/or sufficient conditions for $A+B$ to be invertible

Let $A$ and $B$ be two $n\times n$ real invertible matrices. Are there necessary and/or sufficient conditions (involving only $A$ and $B$ separately, not $(A+B)$ iteself) for $A+B$ to be invertible? ...
1
vote
2answers
27 views

Inverse of the Cross Ratio for Mobius Transformation from Circle to Circle

I'm reading Conway's complex functions of one variable, and in chapter 3 he goes over Cross-Ratios. He defines the cross ratio to be $(z,z_1,z_2,z_3)=\frac{(z-z_3)(z_2-z_4)}{(z-z_4)(z_2-z_3)}$, where ...
1
vote
1answer
137 views

Inversion of the function $ \sqrt x \ln x $

Is there an exact (not asymptotic) inversion of the function $ \sqrt x \ln x $ or can we only obtain this inverse in terms of a power series?
0
votes
3answers
35 views

Finding the inverse of an exponential function

I have this function: $$F_X(x) = \frac{3e^{2x}}{4} + \frac{3e^{4x}}{8} - 0.1$$ of which I am trying to find the inverse function, as in $u = F_X^{-1}(x)$. I made it to this form: $$u= ...
1
vote
1answer
31 views

Linear Algebra - Real Matrix and Invertibility [closed]

Let $M=\begin{pmatrix}A&B\\C&D\end{pmatrix}$ be a real matrix $2n\times 2n$ with $A,B,C,D$ real matrices $n\times n$ that are commutative to each other. Show that $M$ is invertible if and only ...
1
vote
2answers
37 views

What does it mean for f([x])=[2x] for a function mapping R/Z to R/Z?

Let X=R/Z (the circle), with a map $f : X → X$ given by $f([x]) = [2x]$. I'm a little lost on what $f([x]) = [2x]$ means. I thought the function was mapping the equivalence class [x] to the ...
0
votes
2answers
54 views

how to prove that invertible matrix and vectors span the same space?

Given $M$ is an invertible matrix, and {$\vec{v_1}...\vec{v_k}$} spans $R^n$, then {A$\vec{v_1}...A\vec{v_k}$} also spans $R^n$ What does matrix invertibility have to do with span?
1
vote
1answer
50 views

Inverse of linear combination of trigonometric functions [closed]

I have an equation of the form: $$\tan(y)=\alpha_1\cos(x)+\alpha_2\sin(x)$$ where $x$ and $y$ are in $(0,2\pi)$ and the coefficients are real numbers. Implicitly this defines $y$ as a function of ...
1
vote
0answers
40 views

Orthogonal matrix problem

So the question asks: Let $A$ and $B$ be n×nn×n orthogonal matrices, with $n≥2$. Which of the following matrices must be orthogonal? A. $A^TB$ B. The matrix C obtained by multiplying the second ...
5
votes
2answers
196 views

Is there a method to find the inverse of an arbitrary function?

Is it possible to get inverse of all be functions? For example, can we calculate inverse of $y=x^3+x$?
0
votes
1answer
58 views

For any continuous function f(x), how can I split up the function and restrict the domain to find an inverse?

I want to know everything there is to know about inverses for curiosity's sake. I am totally fine finding the inverse of a function where each x maps to a unique y coordinate, but when we get to ...
2
votes
1answer
564 views

How to invert a polynomial function such as $f(x)=x^{\beta}\left((x-1)^6+1\right)$?

I'm trying to replicate a simulation study in a paper. For that I would need the inverse of this function: $f(x)=x^{\beta}\left((x-1)^6+1\right), x\in[0,1]$ Plugging this unto Maxima returns: ...
3
votes
7answers
293 views

How to find the $f^{-1}(x)$ of $f(x)=x^{3}-12x+\frac{48}{x}-\frac{64}{x^{3}}$

It is a question from a quiz. The following is the whole question. Let \begin{eqnarray} \\f(x)=x^{3}-12x+\frac{48}{x}-\frac{64}{x^{3}} , \space x\in (-\infty ,0), \end{eqnarray} find ...
0
votes
1answer
35 views

Show that Ax = b is solvable when [A b] is singular.

I have the following problem: Review: Suppose A is 5 by 4 with rank 4. Show that Ax = b has no solution when, the 5 by 5 matrix [A b] is invertible. Show that Ax = b is solvable when [A b] is ...
1
vote
2answers
35 views

Invertible function that “messes” order [closed]

I am looking for an invertible discrete function $f$ such that given some integer n, if i apply $f(i)$ for $i=0,\dots,n$ I would get all the integers in range $[0..n)$ exactly once, but in a "messy" - ...
0
votes
2answers
24 views

How can I invert the asymptotic form $x^{3/2}=y^{3/2}(1+a/y^2 + … )$ to find $y=y(x)$?

This might sound silly, but the fact there's a $a/y^2$ term in the expansion made me feel a little lost. Could anyone help? Thanks
0
votes
0answers
27 views

Inverse Functions (Discrete Math)

Say you have $f: \mathbb{Z} \to \mathbb{Z}$ defined by $f(x,y) = (2x+y, y)$ How would you check if the function was invertible? As well as determining it's inverse if it is? Thank you
0
votes
1answer
20 views

When left inverse of a function is injective

Consider function $f^{-1}$ which is a left inverse of another function $f$. I require that $f^{-1}$ must be injective. What does it tell me about $f$? In other words, can I put some constraints on $f$ ...
0
votes
2answers
59 views

Inverse of matrices that differ only by one element

Let's suppose that a real matrix $\textbf{A}_{n\times n}$ is nonsingular and its inverse is $\textbf{A}^{-1}_{n\times n}$. Next we change its $A_{ij}$ element to $A_{ij}+a$ and we keep all the other ...
0
votes
2answers
43 views

Inverse of the function $- \log(1-[1-e^{-x^\alpha}]^\beta)$

I have a function as follows, I would like to get the inverse of this function. What is the inverse of $f(x)$? $$ y = f(x) = - \log(1-[1-e^{-x^\alpha}]^\beta)$$ Is my answer correct? $$ f^{-1}(x) = ...
-1
votes
1answer
43 views

Inverse of function arcsin

I'm having trouble finding the solution of the inverse of the function ${\rm f}\left(y\right) = \arcsin\left(\,3 - x^2\,\right)$ Isn't $\arcsin$ the inverse of $\sin$ ?. This is what I have now as ...
1
vote
2answers
58 views

How do I calculate the inverse function of this function?

I have this function: $$ f(x)=\frac{1+\ln(x)}{1-\ln(x)} $$ And i should calculate $f^{-1}(x)$ I am not really sure how to proceed but I think that the first step would be to have x alone, how do I ...
1
vote
2answers
26 views

Need two functions always be composed to prove they are inverses?

Normally, if I claimed that $f: A \rightarrow B$ and $g: B \rightarrow A$ were inverses of each other, I would check for the following results: $f \circ g(b) = b$ and $g \circ f(a) = a$. Suppose I ...
0
votes
0answers
22 views

Calculating the inverse with variables that include logarithm and don't.

I am trying to calculate the inverse of this function and failing. $y_1 =z_1 \sqrt\frac{-2* log(z_1^2 + z_2^2)}{(z_1^2 + z_2^2)}$ Is there a systematic way to go about it?
1
vote
1answer
1k views

Number of flops required to invert a matrix

I have an n-by-n upper triangular matrix $R$ and I can calculate its inverse by back substitution. I cannot make myself see why it needs $O(n^{3}/3)$ flops to do so. Can you explain?
0
votes
1answer
57 views

The inverse of m with respect to n in modular arithmetics

From concrete mathematics problem 4.35. Let $I(m,n)$ be function that satisfies the relation $$ I(m,n)m + I(n,m)n = \gcd(m,n),$$ when $m,n \in \mathbb{Z}^+$ with $m ≠ n$. Thus, $I(m,n) = m'$ and ...
2
votes
2answers
212 views

Calculating an inverse of a split (piecewise defined) function

I am having difficulty taking the inverse of the following function (case-defined): $$ f(x)=\begin{cases} \frac{1}{4 \sqrt{ |1-x|}} & \text{if} \ x\in [0,2] \\ 0 & ...
0
votes
0answers
20 views

Matrix computation of products

If I have two $n \times n$ matrices $A$ and $B$ and a vector $c$,how would I compute the product of $A^{-1}Bc$ ? I know how how to get $A^{-1}$ by doing LU decomposition of $A$ but how do I translate ...
2
votes
1answer
64 views

How do you find the Inverse Laplace transformation for a product of $2$ functions?

If $$\mathscr{L}(y)=\frac{ne^{-pt_0}}{n^2+\omega^2}\left(\frac{1}{p+n}+\frac{n}{p^2+\omega^2}-\frac{p}{p^2+\omega^2}\right)$$ show that $$\bbox[yellow] ...
2
votes
2answers
138 views

How is $X(X^{\prime}X)^{-}X^{\prime}$ symmetric?

For a matrix $X$, a generalized inverse of $X$ is any matrix $Y$ such that $XYX = X$. We use $X^{-}$ to indicate a generalized inverse of $X$. Suppose $X$ is a matrix. $X^{\prime}$ denotes the ...
1
vote
2answers
36 views

Finding the inverse of a matrix given an equation

So I've been given this equation: $A\begin{bmatrix} 2&3&1&5\\ 1&0&3&1\\ 0&2&-3&2\\ 0&2&3&1 \end{bmatrix} = \begin{bmatrix} 0&1&0&0\\ ...
0
votes
4answers
64 views

What is the difference between $f(x)=x^2 +1$ and $f(x)=x^3 -1$ when finding the inverse?

I'm doing some exercises on computing the inverse of each function. In exercise number 56 I did an example where I have to compute the inverse of the function. With my understanding $f(x)=x^2 +1$ ...
14
votes
2answers
790 views

Why aren't integration and differentiation inverses of each other?

Integration is supposed to be the inverse of differentiation, but the integral of the derivative is not equal to the derivative of the integral: $$\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\int ...
-1
votes
2answers
44 views

Matrix invertibility proof? [closed]

Can it be proven that $A^\top A$ is invertible given just the fact that: if $A^\top Ay = \theta$ then $Ay = \theta$? Here $y$ is a vector and $\theta$ is the vector zero.